Questions tagged [rt.representation-theory]

Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.

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How to constructively/combinatorially prove Schur-Weyl duality?

How is Schur-Weyl duality (specifically, the fact that the actions of the group ring $\mathbb{K}\left[ S_{n}\right] $ and the monoid ring $\mathbb{K}\left[ \left( \operatorname*{End}V,\cdot\...
darij grinberg's user avatar
18 votes
2 answers
2k views

Explaining Mukai-Fourier transforms physically

A core concept in mathematics, engineering, and physics is the Fourier Transform (FT) and its many variants (Generalized Fourier Series, Green's Function, Pontryagin duality). The basic algorithm is ...
Tom Copeland's user avatar
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39 votes
7 answers
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Bijection between irreducible representations and conjugacy classes of finite groups

Is there some natural bijection between irreducible representations and conjugacy classes of finite groups (as in case of $S_n$)?
Dan's user avatar
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13 votes
2 answers
970 views

Number of commuting pairs (triples, n-tuples) in GL_n(F_q) (and other groups)?

Question 1 What is the number of pairs of commuting elements in GL_n(F_q) ? I am aware of many results concerning commuting elements in Mat_n(F_q), but I am interested in GL i.e. non-degenerate ...
Alexander Chervov's user avatar
42 votes
3 answers
2k views

Are there "real" vs. "quaternionic" conjugacy classes in finite groups?

The complex irreps of a finite group come in three types: self-dual by a symmetric form, self-dual by a symplectic form, and not self-dual at all. In the first two cases, the character is real-valued, ...
Allen Knutson's user avatar
34 votes
2 answers
2k views

Examples of finite groups with "good" bijection(s) between conjugacy classes and irreducible representations?

For symmetric group conjugacy classes and irreducible representation both are parametrized by Young diagramms, so there is a kind of "good" bijection between the two sets. For general finite groups ...
Alexander Chervov's user avatar
20 votes
1 answer
548 views

$q$-(and other)-analogs for counting index-$n$ subgroups in terms of Homs to $S_n$?

The following formula of astonishing beauty and power (imho): $$ \sum_{n \ge 0} \frac{| \mathrm{Hom}(G,S_n) | }{n! } z^n = \exp\left( \sum_{n \ge 1} \frac{|\text{Index}~n~\text{subgroups of}~ G|}nz^...
Alexander Chervov's user avatar
12 votes
3 answers
1k views

Is there a purely group-theoretic reformulation of an equivalence of subgroups?

There is an equivalence relation between inclusion of finite groups coming from the world of subfactors: Definition: $(H_{1} \subset G_{1}) \sim(H_{2} \subset G_{2})$ if $(R^{G_{1}} \subset R^{H_{1}}...
Sebastien Palcoux's user avatar
51 votes
3 answers
7k views

What to do now that Lusztig's and James' conjectures have been shown to be false?

Lusztig and James provided conjectures for dimensions of simple modules (or decomposition numbers) for algebraic groups and symmetric groups in characteristic $p$. These conjectures have been ...
Chris Bowman's user avatar
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23 votes
2 answers
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Orbit structures of conjugacy class set and irreducible representation set under automorphism group

let G be a finite group. Suppose C is the set of conjugacy classes of G and R is the set of (equivalence classes of) irreducible representations of G over the complex numbers. The automorphism group ...
Vipul Naik's user avatar
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20 votes
3 answers
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Invariants for the exceptional complex simple Lie algebra $F_4$

This is an edited version of the original question taking into account the comments below by Bruce. The original formulation was imprecise. Let $\mathfrak{g}$ denote a complex simple Lie algebra of ...
José Figueroa-O'Farrill's user avatar
15 votes
2 answers
808 views

What are the periodic Dyck paths?

I changed the thread completely so that everything is now elementary linear algebra. A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...,c_n]$ with $c_i -1 \leq c_{i+1}$ for all $i$...
Mare's user avatar
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12 votes
1 answer
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unitary irreps of O(p,q)

I am interested in the irreducible unitary representations of the orthogonal groups $O(p,q)$. By $O(p,q)$ I mean the real Lie groups which preserve the quadratic form of signature $(p,q)$ in $\mathbb{...
Mark Mueller's user avatar
12 votes
1 answer
2k views

A dual version of a theorem of Øystein Ore in group theory

This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved: Theorem: Let $[H, G]$ be a distributive interval of finite groups. Then $\exists g \in G$ such ...
Sebastien Palcoux's user avatar
77 votes
3 answers
9k views

5/8 bound in group theory

The odds of two random elements of a group commuting is the number of conjugacy classes of the group $$ \frac{ \{ (g,h): ghg^{-1}h^{-1} = 1 \} }{ |G|^2} = \frac{c(G)}{|G|}$$ If this number exceeds ...
john mangual's user avatar
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70 votes
10 answers
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Is every finite group a group of "symmetries"?

I was trying to explain finite groups to a non-mathematician, and was falling back on the "they're like symmetries of polyhedra" line. Which made me realize that I didn't know if this was actually ...
Andrew McIntyre's user avatar
51 votes
7 answers
12k views

Why are the characters of the symmetric group integer-valued?

I remember one of my professors mentioning this fact during a class I took a while back, but when I searched my notes (and my textbook) I couldn't find any mention of it, let alone the proof. My best ...
zeb's user avatar
  • 8,533
43 votes
0 answers
2k views

Why are there so few quaternionic representations of simple groups?

Having spent many hours looking through the Atlas of Finite Simple Groups while in Grad school, I recall being rather intrigued by the fact that among the sporadic groups, only one (McLaughlin as I ...
ARupinski's user avatar
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40 votes
5 answers
9k views

Induction and Coinduction of Representations

I'd like to understand the general framework of induction and coinduction of representations. If G is a finite group and H a subgroup, I know that there is a restriction functor from representations ...
Evan Jenkins's user avatar
  • 7,107
40 votes
8 answers
10k views

Ubiquity of the push-pull formula

The push-pull formula appears in several different incarnations. There are, at least, the following: 1) If $f \colon X \to Y$ is a continous map, then for sheaves $\mathcal{F}$ on $X$ and $\mathcal{G}...
Andrea Ferretti's user avatar
31 votes
3 answers
3k views

What's the state of affairs concerning the identification between quantum group reps at root of unity, and positive energy affine Lie algebra reps?

In his paper [1], Finkelberg used Kazhdan-Lusztig's massive work [4,5,6,7,8] to prove that $Rep^{ss}(U_q\mathfrak g)$ (the semisimplification of the category of finite dimensional reps of $U_q\...
André Henriques's user avatar
21 votes
3 answers
6k views

What are the current breakthroughs of Geometric Complexity Theory?

I've read from Wikipedia about Geometric Complexity Theory (GCT) which (if I understood correctly) is a program for coping with the $ P=NP $ problem using algebraic methods. That program seems ...
18 votes
2 answers
1k views

Explicit invariant of tensors nonvanishing on the diagonal

The group $SL_n \times SL_n \times SL_n$ acts naturally on the vector space $\mathbb C^n \otimes \mathbb C^n \otimes \mathbb C^n$ and has a rather large ring of polynomial invariants. The element $$\...
Will Sawin's user avatar
  • 135k
14 votes
2 answers
1k views

Hilbert 90 for algebras

Let $L\diagup K$ be a Galois extension of fields satisfying $\left[L:K\right] < \infty$. Let $B$ be a finite-dimensional (as a $K$-vector space) $K$-algebra. Then, the Galois group $G$ of $L\diagup ...
darij grinberg's user avatar
10 votes
2 answers
1k views

Invariant polynomials for a product of algebraic groups

Let $G_1$ and $G_2$ be connected reductive algebraic groups defined over $\mathbb{C}$ and let $V_1$ and $V_2$ be irreducible representations of $G_1$ and $G_2$ respectively. I'm interested in general ...
Neil's user avatar
  • 103
4 votes
3 answers
2k views

Representation theory of p-groups in particular upper tringular matrices over F_p

Finite p-groups - have p^n elements by definition. According to WP there is rich structure theory. Question: How far is representation theory of p-groups is understood? In case this question is too ...
Alexander Chervov's user avatar
79 votes
12 answers
6k views

Why are characters so well-behaved?

Last year I attended a first course in the representation theory of finite groups, where everything was over C. I was struck, and somewhat puzzled, by the inexplicable perfection of characters as a ...
Saul Glasman's user avatar
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74 votes
7 answers
30k views

A learning roadmap for Representation Theory

As Akhil had great success with his question, I'm going to ask one in a similar vein. So representation theory has kind of an intimidating feel to it for an outsider. Say someone is familiar with ...
67 votes
11 answers
12k views

What is significant about the half-sum of positive roots?

I apologize for the somewhat vague question: there may be multiple answers but I think this is phrased in such a way that precise answers are possible. Let $\mathfrak{g}$ be a semisimple Lie algebra (...
Justin Campbell's user avatar
52 votes
5 answers
7k views

Beautiful descriptions of exceptional groups

I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need ...
zroslav's user avatar
  • 1,412
52 votes
2 answers
5k views

Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?

$\DeclareMathOperator\SO{SO}\newcommand{\R}{\mathbb{R}}\newcommand{\S}{\mathbb{S}}$The periodic table of elements has row lengths $2, 8, 8, 18, 18, 32, \ldots $, i.e., perfect squares doubled. The ...
Eugene Stern's user avatar
49 votes
2 answers
4k views

Which philosophy for reductive groups?

I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the group $G$ we work on is to be ...
Desiderius Severus's user avatar
41 votes
3 answers
3k views

Why is there such a close resemblance between the unitary representation theory of the Virasoro algebra and that of the Temperley-Lieb algebra?

For those who aren't familiar with the Virasoro or Temperley-Lieb algebras, I include some definitions: • The (universal envelopping algebra of the) Virasoro algebra is the $\star$-algebra $...
André Henriques's user avatar
35 votes
4 answers
3k views

How does this relationship between the Catalan numbers and SU(2) generalize?

This is a question, or really more like a cloud of questions, I wanted to ask awhile ago based on this SBS post and this post I wrote inspired by it, except that Math Overflow didn't exist then. As ...
Qiaochu Yuan's user avatar
32 votes
3 answers
3k views

Order of products of elements in symmetric groups

Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying $1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$ whose product has order $...
Stefan Kohl's user avatar
  • 19.5k
32 votes
2 answers
1k views

Analogies supporting heuristic: Weyl groups = algebraic groups over field with one element?

There is well-known heuristic that Weyl groups are reductive algebraic groups over "field with one element". Probably the best known analogy supporting that heuristic is the limit $q\to1$ ...
Alexander Chervov's user avatar
30 votes
3 answers
3k views

Replacing triangulated categories with something better

Gelfand and Manin in their 1988 book on homological algebra write that the non-functoriality of cones means that "something is going wrong in the axioms of a triangulated category. Unfortunately at ...
Hugh Thomas's user avatar
  • 6,075
30 votes
1 answer
2k views

Can one explain Tannaka-Krein duality for a finite-group to ... a computer ? (How to make input for reconstruction to be finite datum?)

Consider a finite group. Tannaka-Krein duality allows to reconstruct the group from the category of its representations and additional structures on it (tensor structure + fiber functor). Somehow ...
Alexander Chervov's user avatar
26 votes
2 answers
6k views

Which finite groups have faithful complex irreducible representations?

Obvious necessary condition is that the center must be a cyclic group. Is it sufficient (doubt here)? If not, is there any nice characterization in terms of group structure, without appealing to ...
Fedor Petrov's user avatar
22 votes
1 answer
1k views

Weyl group actions on 0-weight spaces

For a complex simple Lie group G with a maximal torus T, we can take a highest-weight representation V of G and look at the 0-weight space, i.e. the subspace of V of elements invariant under T. This ...
Matthew Tai's user avatar
19 votes
2 answers
2k views

Dual versions of "folding" symmetric ADE Dynkin diagrams?

Start with the Dynkin diagram of an irreducible root system, typically associated with a simple Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE diagrams ...
Jim Humphreys's user avatar
18 votes
6 answers
3k views

Reference request: representation theory of the hyperoctahedral group

I was wondering if someone knows a good reference for the representation theory of the hyper-octahedral group $G$. The hyper-octahedral group $G$ is defined as the wreath product of $C_2$ (cyclic ...
Puraṭci Vinnani's user avatar
17 votes
0 answers
678 views

Monstrous Langlands-McKay or what is bijection between conjugacy classes and irreducible representation for sporadic simple groups?

Context: The number of conjugacy classes equals to the number of irreducuble representations (over C) for any finite group. Moreover for the symmetric group and some other groups there is "good ...
Alexander Chervov's user avatar
17 votes
5 answers
2k views

Good source for representation of GL(n) over finite fields?

I'd like to gain some understanding of unitary representations of GL(n) over finite fields. Any good source would be appreciated. ======== edit ========= My original question was ambiguous. ...
user1258240's user avatar
15 votes
3 answers
2k views

Integration of a function over 7-sphere

Suppose we have $x_1^2 + y_1^2 + x_2^2 + y_2^2 + x_3^2 + y_3^2 + x_4^2 + y_4^2 = 1$ and we define $z_j = x_j + iy_j$, where $j = 1,\,2,\,3,\,4$. The problem is finding or approximating the ...
Hrushikesh Pawar's user avatar
15 votes
3 answers
4k views

Generators of invariant polynomials of semisimple Lie algebra

Suppose $\mathfrak{g}$ is a complex semi-simple Lie algebra. By a theorem of Chevalley, we know that $S(\mathfrak{g})^\mathfrak{g}$, i.e. the $\mathfrak{g}$ invariant polynomials, is generated by $l$ ...
Qijun Tan's user avatar
  • 577
13 votes
1 answer
711 views

A maximal element, where Schur gives a minimal element

Let me recall a result due to I. Schur, which I learnt from F. Goldberg's answer to my MO question Hadamard-like inequalites for positive definite symmetric matrices. If $H$ is a subgroup of $\frak ...
Denis Serre's user avatar
  • 51.5k
13 votes
0 answers
719 views

Bijection between conjugacy classes and irreducible representation of Weyl group = Langlands correspondence over "field with one element"

Context: The number of conjugacy classes equals to the number of irreducuble representations (over C) for any finite group. Moreover for the symmetric group there is well-known "natural bijection" ...
Alexander Chervov's user avatar
13 votes
3 answers
1k views

Characterization of Frobenius complements

I have learned that Frobenius complements are characterized (among finite groups) by having a fixed point free complex representation. That is, a finite group $G$ is a Frobenius complement if and only ...
Joonas Ilmavirta's user avatar
12 votes
1 answer
5k views

What are tame and wild hereditary algebras?

What are tame and wild hereditary algebras? Are they related to hereditary rings? (Those are rings for which every left (resp. right) ideal is projective, equivalently, for which every left (resp. ...
Jose Brox's user avatar
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